In the paper we study the existence of solutions of the random differential inclusion x˙tG(t,xt)     P.1,t[0,T]-a.e.x0=dμ, where G is a given set-valued mapping value in the space Kn of all nonempty, compact and convex subsets of the space n, and μ is some probability measure on the Borel σ-algebra in n. Under certain restrictions imposed on F and μ, we obtain weak solutions of problem (I), where the initial condition requires that the solution of (I) has a given distribution at time t=0.